Topological Vector Space Valued Measures on Topological Spaces

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Authors

  • Surjit Singh Khurana
     The University of Iowa, Department of Mathematics, Iowa City, IA ,US

DOI:

https://doi.org/10.18311/jims/2019/21590

Keywords:

vector measures, measure representation of linear operators, Alexandrov's theorem
46E10, 28C05, 28C15, 46G10, 28B05.

Abstract

If X is a compact Hausdorff space space, E is a complete Hausdorff topological vector space and μ : (C(X),ll.ll) → E a linear continuous exhaustive mapping, we rst give a different proof that there is then a unique reqular, L∞-bounded, exhaustive E-valued Borel measure μ on X such that μ(f) = ∫ fdμ, ∀f ∈ C(X). Then we consider X to be a completely regular Hausdorff space and prove the extension of Alexanderov's theorem: X is a completely regular Hausdorff space and μ : Cb(X) → E a linear, continuos, exhaustive mapping and F is the algebra generated by zero-sets in X. Then there exist a unique nitely additive, exhaustive measure ν : F → E such that (i) ν is L∞-bounded i.e. the absolute convex hull of ν(F) (Γ(ν(F))) is bounded in E; (ii) ν is inner regular by zero-sets and outer regular by positive-sets; (iii) âˆ« fdν = µ(f), ∀f ∈ Cb(X).

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Published

2019-08-22

How to Cite

Singh Khurana, S. (2019). Topological Vector Space Valued Measures on Topological Spaces. The Journal of the Indian Mathematical Society, 86(3-4), 259–271. https://doi.org/10.18311/jims/2019/21590

Issue

Volume 86, Issue 3-4, July-December 2019

Section

Articles

License

Copyright (c) 2019 Surjit Singh Khurana

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

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Received 2018-07-19
Accepted 2019-03-20
Published 2019-08-22

 

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